Richard Goldstein wrote:
> I have just returned from Salt Lake City to find this
> interesting discussion. Although I have nothing to add
> for the situation where the identity link is used, I
> note that others have complained about failure to
> converge when using the log link (relative risk
> interpretation). For those cases, there is a solution:
> use poisson regression with robust error variance;
> see Zou, G (2004), "A Modified Poisson regression
> approach to prospective studies with binary data,"
> _American Journal of Epidemiology_, 159: 702-706.
>
> Rich
Thanks for reminding me (and the list) about that Rich. And slightly
belated thanks to Bobby for posting his code for the loglogit link (i
wasn't expecting a response until after the weekend!)
A quick comparison (below) suggests that Bobby's "loglogit" link may be
a bit more efficient than poisson with robust SEs, in the same way that
Spiegelman & Hertzmark (2005) suggest the log link should be more
efficient when it converges. See also Peterson & Deddens (2005) (and
refs therein), who give yet another approach that no-one's implemented
in Stata, AFAIK (can't say i find it very attractive myself). Comparing
them all sounds like an undergrad stats project... One obvious advantage
of Bobby's approach is that likelihood ratio tests remain available.
Spiegelman D, Hertzmark E. Easy SAS Calculations for Risk or Prevalence
Ratios and Differences. Am.J.Epidemiol. 2005;162:199-200.
<http://dx.doi.org/10.1093/aje/kwi188>
Petersen MR, Deddens JA. RE: "EASY SAS CALCULATIONS FOR RISK OR
PREVALENCE RATIOS AND DIFFERENCES". Am.J.Epidemiol. 2006;163:1158-1159.
<http://dx.doi.org/10.1093/aje/kwj162>
poisson for price mpg, robust nolog
Poisson regression Number of obs
= 74
Wald chi2(2) =
27.23
Prob > chi2 =
0.0000
Log pseudolikelihood = -43.959394 Pseudo R2 =
0.0971
------------------------------------------------------------------------------
| Robust
foreign | Coef. Std. Err. z P>|z| [95% Conf.
Interval]
-------------+----------------------------------------------------------------
price | .0001289 .0000593 2.17 0.030 .0000127
0002451
mpg | .1052061 .0201699 5.22 0.000 .0656738
1447385
_cons | -4.427599 .7772089 -5.70 0.000 -5.950901
-2.904298
------------------------------------------------------------------------------
glm for price mpg, fam(bin) link(loglogit) nolog
Generalized linear models No. of obs
= 74
Optimization : ML Residual df
= 71
Scale parameter
= 1
Deviance = 75.68075809 (1/df) Deviance =
1.065926
Pearson = 66.85115827 (1/df) Pearson =
9415656
Variance function: V(u) = u*(1-u) [Bernoulli]
Link function : g(u) = ln(u) -> logit(u) [Log to 0.99, then logit]
AIC =
1.103794
Log likelihood = -37.84037905 BIC =
-229.9079
------------------------------------------------------------------------------
| OIM
foreign | Coef. Std. Err. z P>|z| [95% Conf.
Interval]
-------------+----------------------------------------------------------------
price | .0001015 .0000442 2.30 0.022 .000015
000188
mpg | .1051885 .0256521 4.10 0.000 .0549112
1554657
_cons | -4.246603 .8992546 -4.72 0.000 -6.009109
-2.484096
------------------------------------------------------------------------------
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